FAQs

• Does BubbleDet include the zero mode factors automatically?

  Yes! BubbleDet’s function findDeterminat() includes all zero mode factors.

• Should I use the tree-level potential or the one-loop effective potential in BubbleDet?

  For simple cases, such as were considered in Callan and Coleman’s classic work[1] and in our introduction, the functional determinant arises from a saddle-point approximation of the path integral. So, the tree-level potential and not the one-loop potential should be used within both the bounce equations and the functional determinant. Using the one-loop effective potential would amount to double counting fluctuations, as well as an uncontrolled derivative expansion.

  However, when loop corrections are of leading-order importance for the potential, such as in thermal or radiatively-induced transitions, this question is more subtle[2]. In such cases, the relevant potential for both the bounce equation and the functional determinant is the tree-level potential for the nucleation scale effective field theory[3]. The effective field theory approach avoids double counting and uncontrolled derivative expansions, and yields results which are real, gauge invariant and renormalisation scale invariant order-by-order.

• For a thermal cosmological phase transition, what dimension should I use in BubbleDet?

  In BubbleDet, the dimension \(d\) should match the \(O(d)\) symmetry of the background bubble. So, for a vacuum transition the dimension should be \(d=4\), while for a thermal transition the dimension should be \(d=3\).

• What does the thermal flag do exactly?

  Setting the thermal flag equal to True adds the dynamical prefactor, in the following approximation[4]:

  \[\begin{split}&\texttt{findDeterminant(thermal=True)}= \\ &\qquad\qquad \texttt{findDeterminant(thermal=False)} - \log\bigg(\frac{\sqrt{|\lambda_-|}}{2\pi} \bigg)\end{split}\]

  where \(\lambda_-\) is the negative eigenvalue of the fluctuations around the bubble. For more information on the dynamical prefactor, and the approximation we have adopted for it, see the BubbleDet paper[5].

References

[1]

Curtis G. Callan, Jr. and Sidney R. Coleman. The Fate of the False Vacuum. 2. First Quantum Corrections. Phys. Rev. D, 16:1762–1768, 1977. doi:10.1103/PhysRevD.16.1762.

[2]

Erick J. Weinberg. Vacuum decay in theories with symmetry breaking by radiative corrections. Phys. Rev. D, 47:4614–4627, 1993. arXiv:hep-ph/9211314, doi:10.1103/PhysRevD.47.4614.

[3]

Oliver Gould and Joonas Hirvonen. Effective field theory approach to thermal bubble nucleation. Phys. Rev. D, 104(9):096015, 2021. arXiv:2108.04377, doi:10.1103/PhysRevD.104.096015.

[4]

Ian Affleck. Quantum Statistical Metastability. Phys. Rev. Lett., 46:388, 1981. doi:10.1103/PhysRevLett.46.388.

[5]

Andreas Ekstedt, Oliver Gould, and Joonas Hirvonen. BubbleDet: A Python package to compute functional determinants for bubble nucleation. arXiv, 8 2023. arXiv:2308.15652.